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Research on routine problem solving (e.g. the typical story ,. problem) was reviewed to facilitate the identification and dissemination of promising practices for teaching routine problem solving, and to provide suggestions and directions for further research in the area. Promising teaching practices which were identified included giving attention to processes involved in solving routine problems (e.g., write an equation, make a chart) and devoting time to developing the meanings of mathematical vocabulary and symbols. Areas identitied as warranting further research included studies that examine the role of language variables (both syntactic and semantic) in the odecodingn pbase of solving a routine problem. Appendix C contains a coded bibliography which may be of great value to researchers in problem solving. (

Sowder, Larry: And Others

A Review of Research on Solving Routine Problems in

Pre-College Mathematics

Northern Illinois Univ., De Kalb.

National Science Foundation, Washingtnn,

79NSP-SED-77-1915799p.: Not available in hard copy due to marginallegibility of original document

MF01 Plus Postage PC Not Available from EDRS.

Algebra: *Bibliographies: *Educational Research;

Elementary Secondary Education: Information

Dissemination: Mathematical Vocabulary; MathematicsCurriculum: *Mathematics Education; *MathematicsInstruction: Memory: *Problem Solving; *ResearchReviews (Publications): Symbols (Mathematics)

ABSTRACT

Research on routine problem solving (e.g the typical

"story ,. problem) was reviewed to facilitate the identification and

dissemination of promising practices for teaching routine problem

solving, and to provide suggestions and directions for further

research in the area Promising teaching practices which were

identified included giving attention to processes involved in solving

routine problems (e.g., write an equation, make a chart) and devotingtime to developing the meanings of mathematical vocabulary and

symbols Areas identitied as warranting further research included

studies that examine the role of language variables (both syntactic

and semantic) in the odecodingn pbase of solving a routine problem

Appendix C contains a coded bibliography which may be of great value

to researchers in problem solving. (MK)

***********************************************************************

Reproductions suppl3ed by EDPS are the best that can be made

from the original document

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NATIONAL INSTITUTE OF EDUCATION pets DOCUMENT HAS BEEN PERRO-

THE PERSON OR ORGANIZATION ORIGIN AT:Nc IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECESSARILY REPRE.

EDuCTtON POSITION OR POLICY

A REVIEW OF RESEARCH ON SOLVING ROUTINE PRdBLEMS

IN PRE-COLLEGE MATHEMATICS

Larry Sowder Jeffrey C Barnett Kenneth E Vos

1979

$ UMMARY

This project reviewed the research on routine problem solving (e.g., the

typica- "story" problem) with an aim toward (a) identifying and disseminating

promising practices for teaching routine problem solving and (b) sug3esting

dftections for further research in the area The investigators surveyed

relevant dissertations, journal articles, and files of research studies.

Products of the work include a chapter in a National Council of Teachers of

Mathematics yearbook oriented toward teachers, and chapters in two monographs

oriented toward educational researchers Various talks at meetings for teachers

and researchers have also been scheduled.

Practices which might improve the teaching of routine problem solving

include these;

1 Give attention to processes involved in solving routine problems (e.g.)

write an equatlon, make a chart).

2 Devote time to developing the meanings of mathematical vocabulary

and symbols.

3 Teach that reading of a mathematical problem is different from

reading less technical prose, and requires multiple readings with

attention to vocabulary and relationships among variables.

4 Have the learners make up, and solve, their own word problems.

Ar2as in which further research and development seem warranted include

these:

5 Instrumentation is needed for process-analysis studies, both for

protocol coding and process measurement.

6 Studies that examine the role of language variables (both syntactic

and semantic) in the "decoding" phase of solving a routine problem

should contribute to our knowledge of teaching problem solving.

7 Whether different ways of presenting problemsobjects, pictures,

words help children of different ages and mental characteristics

needs examiaation.

Narrative

Appendix A: A Review of Selected Literature

on the Role of Memory in Arithmetic and Algebra Word Problem Solving

Appendix B: Publications Based on the Project

A Review of Research on Solving Routine Problems

in Pre-College Mathematics

ObitctIm

The aims of the project were as follows:

1 Review and evaluate research on routine problem solving in college mathematics.

pre-2 To identify directions for further research on routine problem

solving in mathematics.

3 To make the findings about routine problei solving available toclassroom teachers and researchers.

"0; eL Lee terms in the objectives do not have standard meanings. Here

are explanations of how the terms were used in the project:

Problem A problem is a task which does not immediately suggest to

the solver a systematic procedure for resolving the task

(i.e., an algorithm). Thus, a person's kriowledge ofalgorithms determines whether a given task is a problem for that person Multiplying with multi-digit numerals would be

a problem for third graders but not a problem for most seventh graders.

Routine vs nonroutine The routineness of a problem mays in a rough

way, be defined by the nature of its solution. The solution

of a now7outine pmblem requires considerable analysis,

synthesis, and perhaps some novelty of approach. On theother hand, the solution of a routine problem requires only

a relatively small amount of analysis and no unusual insights.The typical verbal problem in pre-college mathematics books

is an example of a routine problem. Such problems usuallyrequire only the selection of an appropriate computation.

In elementary school book, protlems described as "challenges"

or "brainbusters," are ueually nonroutine problems.

Rationale

The ratanale for the project was based on four things. First, attention

to routine problems in mathematics is important. Effective citizenship as a

consumer, as a wage earner, as a taxpayer, requires an ability to solve a

myr.Lad of routine problems Checking purchases, calculating interest costs,

evaluating budgets, determining best buys, planning meals all these are

sample routine problems.

Second, unfortunately many students.do not solve problems well The

availability of handheld calculators is of no value if a person does not

know which buttons to press It is well known that students are not usually

fond of "story" problems, and the firdt National Assessment of Educational

Progress (1975) offered evidence that people are not very proficient in solving

routitie problems If existing research evidence suggests that certain

procedures for teaching routine problem solving are promising, these should

he identified and disseminated.

Third, at least a few such promising practices are identified in the

literature but do not appear to be widely known For example, VanderLinde (1.964,

found a positive effect on problem solving from spending time on developing

1

meanings for symbols and on studying quantitative vocabulary.

-Finally, directions for further research on routine problem solving in

mathematics might be identified by an analysis and critique of the existing

research.

Procedures The investigators Barnett, Vos, and Sowder identified as many studies

of routine Problem solving in pre-college mathematics as they could, through

searches of dissertation abntracts and ERIC files and through examination

of journals deemed most likely to contain such studies. The studies were

categorized with an adaptation of Kilpatrick's taxonomy of variables in

problem solving research (1978) The most promising dissertations, reports,

and articles were studied in full. On the basis of this wo-A, manuscripts

were prepared (see Products below) In addition, Dr Edward Silver agreed

to consult with the project and prepared a project paper on memory aspects

of routine problem solving (see Appendix A).

Limitations The project was necessarily limited in scope. Which journals

shouli be conceutrated on? llow far afield from routine problems in mathematics

should we explore? The homely when-do-we-stop-reading-and-start-writing

question demanded an answer, dictated by manuscript deadlines in our case.

Hence, we cannot claim to have accomplished a comprehensive search. For

example, the information-processing approach to routine problem solving was

slighted Work along these lines is currently popular, perhaps too current

for a dispassionate critique or for an even moderately thorough survey.

Silver's project paper (Appendix A) does draw on work in information processing,

however Anotherattractive but unexplored body of work was in problem

solving in fields other than mathematics e.g., science, therapy, busineas.

It may well be that important implications fcr mathematics education lie in

studies in such domains.

Products

Serendipitously, the dissemination objective of the project was realized

through the post-proposal appearance of plans for two works on problem

solving a monogrsolving aph on "solving applied" problem solving (R Lesh &solving amp; D Mierkiewicz, Eds.)

and a National Council of Teachers of Mathematics yearbook on problem

solving (S Krulik, Ed.) Proposals and drafts of chapters for these two

forth-coming works were prepared and accepted. In addition, Barnett prepared a

chapter for a mono6rupn on task variables in mathematical problem solving

(G Goldin & E McClintock, Eds.), soon to appear. Citations for these

publications are summarized in Appendix B. The intent to reach the teacher

audience through the Arithmetic Teacher was abandoned on the appearance in

the November, 1977, issue of an excellent problem-solving article al3ng the

lines planned (Suydam & Weaver); the yearbook chapter served as a

teacher-oriented article.

Presentations based on the project have aided, or will aid, in the

dissemination of the major findings: Barnett (state meeting of the Illinois

Council of Teachers of Mathematics, October, 1978); Sowder (regional Illinois

meeting, March, 1979; National (ouncil of Teachers of Mathematics regional

meeting, March, 1980); Vos (California Mathematics Council, Southern Section,

Novembei, 1979); Barnett, Vos, and Sowder (National Council of Teachers of

Mathematics national meetinc April, 1980).

Finally, one product of the*project was not planned in the original

pro-ject but may be of great value to researchers'in problem solving: the

coded bibliography (see Appendix C) This bibliography will be made available

to interested researchers through contacts in the Special Interest Group on

Research in Mathematics Education of the American Educational Research

Associ-ation, and through ERIC.

Selected research recommendations

The following condensed excerpts from the manuscript for the Lesh monograph

represent the flavor of our recommendations for further research in routine

problem solving in mathematics:

1 For studies of the processes involved in routine problem solving,

test instruments that emphasize such process variables must be developed, as well as a protocol scoring-coding scheme that is both elegant and efficient.

2 Studies that attempt to determine the role of syntax and semantic

variables in the decoding process in the first stage of problem solving are of particular importunce.

3 Another area of needed research is that concerned with the ment of instruction in reading and its relationship to improved problem solving ability.

improve-4 Although the linear regression model has shown some promise as a

research technique in the area of language variables and routine problem solving, it is clear that in its present form it falls short

of being able to predict problem solving success. Improvements inthe model might include different criteria of importance; it would

be helpful for studies to provide data on several dependent measures used with several measures of importance.

5 The relative effects of different formats words, pictures, for problems should be investigated, particularly as they relate

objects to learner characteristics.

6 What within-format variations make a difference?

7 Studies with positive results should be replicatedfor example,

Keil's 1964 study, in which student-generated and student-solved problemi apparently led to improved problem solving.

e Would concentrated attention'to routine problems give the same

striking results as in Bramhall's 1939 study (8 months growth in 2.5 months)?

9, Cooperation among researchers interested in routine problem solving must increase so that common problemst similar instruments, and shared data analysiq can be more easily facilitated.

References

Bramhall, k, W. An experimental study of two types of arithmetic problems.

The alOUrnal of Experimental Education, 1939, 8, 36-38.

Goldin, G A & McClintock, C E (Eds.) Task variables in mathematical

_problem solving Columbus, Ohio: ERICISMEAC, 1979 (in press).

Keil, G E Writing and solving original problems as a means of improving

verbal arithmetic problem solving ability (Doctoral dissertation,

Indiana University, 1964). Dtusrl,asjoruatuttii, 1965, 25, 7109-7110.(University Microfilms No 65-2376)

Kilpatrick, J. Variables and methodologies in research on problem solving.

In L Hatfield (Ed.), liathettobler, Columbus, Ohio:

ERIC/SMEAC, 1978.

Krulik, S Problem solving Reston, Virginia: National Council of Tc:achers

of Mathematics, in press.

Lesh, R., & Mierkiewicz, D Applied problem solving Columbus, Ohio: ERIC/

SMEAC, 1979 (in press).

National Assessment of Educational Progress. Consumer math, selected results

from the First National Assessment of Mathematics. Mathematics Report

No 04-MA-02 June, 1975.

Suydam, M N., & Weaver, J F Research on problem solving: Implications

for elementary school classroon,s Arithmetic Teacher, 1977, 25(2),

40-42.

VanderLinde, L F Does the study of quantitative vocabulary improve

problem-solving? Elementary School Journal, 1964, 65, 143-152.

Project Paper

A Review of Selected Literature

on the Role of Memory in

Arithmetic and Algebra Word Problem Solving

-by

Edward A Silver

San Diego State University

When I recently mentioned to a colleague that I was writing

a paper on the role of memory in solving mathematics problems,

she remarked.: "Well, it certainly helpsi", The fact is, however,

that there are three distinct ways in which memory might interactwith problem-solving performance First, information from

previous problems might not be available either because the

solver has not encoded it or because it has been encoded in a

fashion that makes it difficult or impossible to retrieve. In

this case, memory would have little or no ,effect on

problem-solving.performance The second way in which memory

might interact with problem solving is to have a negative

effect Gestalt psychologists (e.g., Duncker, 1945; Luchins,

1942; Wertheimer, 1959) have examined extensively the instances

in which past experience can negatively affect present

problem-solving performance The third way, of course, is the one

to which my colleague was referring, in which information

gained from previous problem-solving encounters is successfully

recalled and used to solve a new problem.

Psychologists have studied the role of memory in general

problem-solving activities, and the classical theories have differedgreatly in the importance given to memory. The.present review

has been greatly influenced by the modern information-processing

view of problem-solving (Newell & Simon, 1972). The reader

who is unfamiliar with information processing.psychology can

find excellent discussions in Mayer (1977),or Norman (1976).

The human information-processing model ean be divided

into two general components: perception and mAmOry. Memory

is described in the literature in a variety of ways using variousmodels and analogies It is not possible in this paper to

summarize the various models, but the reader %ill find an

excellent summary of hypothesized memory structures and theories

in Gagne and White (1!-0,8) and a briefer, but highly

readable, summary in Shavelson and Porton (Note 1).

In writing this paper, I have attempted not to Auplicatethe work of Greono (1973), in which he applied the general

information-processing view of problem solving in reviewing

studies relating memory and problem solving Therefore, this

review has generally confined its attention-to studies conductedsince Greeno's excellent review and to studies that deal in

some way with the concerns ,c4 mathematical word problem solving.While no claims are made for completeness of the review, it is

hoped that the reader will become acquainted with the dominant

theories, majets-teTults, and possible future directions for

research on the role of memory in solving mahematics word

problems

Arithmetic Problem Solving

Although the arithmetic problem-solving competence'of childrenhas been of great interest to researchersi their primary focus hasbeen the product (i.e correct/incorrect answer) rather than the

process In their recent work, Jim Greeno, Joan Heller, and maryRiley (e.g c,eeno, Note 3; Heller, Note 4; Heller & Greeno, Note5; Riley, Note 6; Riley & Greeno, Note 7) have applied the

information-processing viewpoint to arithmetic problem solving

of Wisconsin suggests that the Heller-Greeno model is both

incomplete and partially incorrect (see Carpenter, Hiebert, &

Moser, Note 8; Carpenter & Moser, Note 9), the model deserves

careful consideration here since it points to the important

iole of memory in the solution of arithmetic word problems,

especially by young children

In the Heller-Greeno model, initial understanding of

a problem is viewed as a process of constructing an integrated

semantic representation of the general quantitative relations

in the problem situation Subsequent selection of the correct

operation is based on a direct association between this semanticrepresentation (corresponding to one of three fundamental

schemata in the Heller-Greeno model) and the operators (availableand associated with the given schema). Carpenter and Moser

(Note 9) have suggested that there are more than three fundamentalschemata and that children do not appear to reduce all problems

to instances of a particular type and apply a single strategy.

Nonetheless, their work suggests the fundamental importance ofsemantic processing in arithmetic word problem solving.

The Heller-Greeno characterization is especially interestingbecause it contrasts with the earlier information-processing

model for word problem solving proposed by Bobrow (1968), in

which the problem text is interpreted phrase-by-phrase, using

syntactic function tagging, and directly transformed into an

equation or system of several simultaneous equations representingthe problem situation Support for Heller & Greeno's

emphasis on semantic processing may be found in studies that

collectively suggest that the ability to represelit a word

problem in the form of an equation or a system of equations

is not a necessary condition for successful solution of the

problem For example, several studies (Buckingham & Maclatchy,

1930; Carpenter, Hiebert, & Moser, Note 8) have found that

young children can correctly solve some word problems before

receiving any formal instruction in equation writing or the

translation process Furthermore, Riley and Greeno (Note 7)

reported that second-grade children sometimes found it difficult

or impossible to write equations for problems they had already

solved Additional support for the importance of semantic

processing comes from reports of successful problem solvers

and their characteristics (e.g Larkin, Note 10; Paige & Simon,

1966; Simon & Simon, 1978); discussion of these reports is

found later in this paper

Therefore, the data suggest that the crucial understandings inthe process of solving a problem are those involving "making sense"

of the problem situation; i.e applying to the problem at hand

real world or technical domain-specific semantic knowledge that is

stored in LTM The typical instruction given to students who arelearning to solve word problems usually encourages such semanticprocessing, but the usual emphasis is on syntactic procedural

mechanisms

4

For example, two reasonably well known procedures taught

to children are the "Wanted-Given" approach and the

"Action-Sequence" approach (Wilson, 1964). Both approaches emphasize

a certain amount of semantic processing, in that students are

trained to "look for" the wanted-given relationship or the

imagined action-sequence embedded in a problem Nevertheless,the major emphasis of instruction in either procedure is on thecomposition of an equation, often in a rather rote fashion thatseems somewhat independent of the initial semantic processingthat is presumed to occur

Unfortunately, at this time, we know very little about

how children "see" word problems. For example, what is it thatsuggests that a given problem is a subtraction problem, and

how is that realization associated with the production of an

appropriate equation or operational sequence?

One particularly fruitful line of research would appear to

be the identification of the fundamental units of children's

understanding of arithmetic concepts and problems. The work ofCarpenter and his associates is noteworthy in this regard.

Another approach is being taken by Alan Rudnitsky at Smith

College Rudnitsky has been interviewing children to determine

the "primitives" (basic elements) of their arithmetic schemata.Such work can be seen as extending the seminal studies of

Erlwanger (1975) and Ginsburg (1977) on children's underStanding

of arithmetic concepts and principles.

Algebra Problem Solving

If a competition were held to determine the most influentialAnd popular memory construct in the area of algebra problem

solving, there is no doubt that the current winner would be

the notion of "schema" (taken here to be equivalent to notions

such as "frame" or "script") A memory schema, as it is

typically conceptualized today, is a cluster of

knowledge-concepts, procedures, and relations'among these - that defines

a more complex and frequently encountered concept or

phenom-enon.

Schemata have been variously defined and discussed in the

current literature on memory models (e.g., Bobrow & Norman,

1975; Rumelhart & Ortony, 1977), but certain common properties

are invariant across the different definitions For example,

a schema represents a prototypical abstraction of a complex

concept, and the schema is derived from past experience with

numeruus exemplars of the complex concept Furthermore, a

schema can guide the organization of incoming information into

clusters of knowledge that are "instantiations" of the schema

(Thorndyke & Hayes-Roth, 1979) The notion of schema was

first proposed in connection with algebra word problems by

Hinsley, Hayes, and Simon (1977) and has been recently adopted

by Bob Davis and his colleagues (Davis, Note 11; Davis, Jockusch,

& McKnight, 1978) in discussing algebra problem solving in

general

Hinsley, et al found that their subjects used two differentprocedures in solving algebra word problems One approach involved

a line-by-line direct translation procedure, such as the one

proposed by Bobrow (1968i and discussed previously The second

approach involved reading the entire problem before formulating

any equations or writing any relations among lariables. This

second approach - the "schema" approach - emphasized the fundamentalimportance of semantic knowledge and major decisions occurring

early in the comprehension process The data provided by Hinsley,

et al. demonstrate that the "schema" approach is typically used

by successful solvers and that the line-by-line procedure is a

default process used only if the problem is not successfully

matched to one of the solver's available problem category schemata

Since the Hinsley, et al study, further evidence of the

existence of problem category schemata has been produced involvingalgebraically naive subjects (Silver, 1977; SilversNote 12;

Silver, Note 13), college students solving physics problems

(Larkin, Note 10), and a wide variety of mathematical tcpics

and students of various ages (Davis, Jockusch, & McKnight, 1978).The results of these studies suggest that problem schemata not

only exist but are used by successful problem solvers in planningtheir approach to solving a given problem.

Larkin (Note 10) analyzed the protocols of college studentssolving rather complex physics problems. She found evidence thatsuccessful problem solvers performed an initial "qualitative

analysis" before writing any equations In the early stages

of a problem solution, saccessful solvers constructed

represen-tations of the physical situation described in the problem,

and they subsequently modified and elaborated the representation

., by ilicluding supplementary information required for a completeundertanding of the problem situation but not given explicitly

in the problem's written statement

Larkin's protocols provide evidence that successful solversretrieve from memory preliminary "chunkE" or "schemata" of

related physics concepts and principles and apply the "chunks"

to some aspect of their problem representation Problem featuresare elaborated further if necessary in relation to the "chunk"

under consideration as the solver attempts to determine the

applicability of the knowledge cluster to the problem

repre-sentation or the solver exits from the problem solution episode.

Upon finding a ,"chunk" that adequately "fits" the problem

representation, the solver generates a solution procedure

The findings of Hinsley, et al and Larkin suggest that

problem schemata exist and may play a critical role in solvingcertain classes of problems, such as algebra word problems

Fa7 less is known about the mechanisms of schema construction;

i.e how students form problem schemata

Research conducted by Krutetskii (1976), Chartoff (1977),and Silver (1977) has suggested several dimensions along whichstudents might form schemata Silver asked eighth grade

students to sort a set of word problems into groups of problemsthat were "mathematically related"; Chartoff asked students

to rate problem pairs on a continuous scale, ranging from

extremely dissimilar to extremely similar The_two investigatorsindependently identified three similarity dimensions perceived

by the students: mathematical structure, contextual (cover story)details, and the nature of the question asked In addition,

9Chartoff found that students could recognize generalizations

and specializations, and Silver identified a tendency to form

clusters of problems on the basis of a common measurable

quantity, such as age or.weight

The findings of Chartoff and of Silver, together with the

observation by Krutetskii that good problem solvers tend to

notice and recall a problem's structure, whereas poor solv

notice and recall only the details of a problem's statement,

suggest that students apprehend the important aspects of a

problem in different ways This initial processing is clearly

influenced by existing problem schemata, if any exist for the

solver, and form the basis for construction of new schemata

Recent work by Silver (Note 12, Note 13) suggests that students

cluster recall of problem information abound existing schemata,

that they use information from previously solved problems when

solving what they perceive to be related problems, and that

good and poor problem solvers exhibit qualitatively different

clustering and recall performances These findings will be

discussed in more detail in a later section of the paper

Whereas the investigations cited above involved no direct

schema-forming instruction, it is common for algebra word problem

instruction to organize problems into "types"; such as "age"

problems, "mixture" problems, and "work" problems. The emphasis

on "types" may lead to the students' forming problem schemata on

the basis of those categories Hinsley, et al found that their

college subjects did organize algebra word problems into groups

that conformed to the stereotypic groupings typically taught to

first year algebra students Nevertheless, it is evident that

9

ta not all students who receive the same instruction form the same

problem schemata

Instruction involving problem "types" was prevalent in the SovietUnion in the 1930s and 1940s The usual pedagogical style involvedteaching students to identify problem "types", to recall "model"

solutions and to ignore the influence of unfamiliar settings or

extraneous data Russian school psychologists thus had an

oppor-tunity to study the process by which a student forms the concept

of a problem type Although their paradigms differ from the moderninformation-processing viewpoint, their findings are germane.

Kalmykova (1947/1969) reported that the extensive use of modelproblems tends to reduce the act of problem solving to a choice of

conditions of the problem Menchlnskaya (1946/1969) also

expressed the view that typification leads students to search

their memories for models to "fit" the given problem. She

reported that such instruction led students to search their

memories to 'reconstruct a previously encountered problem to

serve as a model, rather than examine the problem's conditions

effort to construct an appropriate solution.

It woul,d appear that schemata are important especially in theformulation of problems in which the contextual details, the

semantics of the cover story, match the underlying problem

structure in an expected way For these problems, if the necessaryschema is available to the solver, then a solution may be obtained;otherwise: the line-by-line default procedure must be used

The data of Chartoff (1977), Krutetskii (1976), and Silver (1977)suggest that schemata might be formed along inefficient dimensions;i.e with respect to non-structural problem characteristics. The

reports of Kalmykova (1947/1969) and Menchinskaya (1946/1969)

suggest that, even if problem schemata are formed along).the

appropriate dimension of mathematical structure, they may not be

useful in solving a problem when the solver fails to

analyze carefully the conditions of the problem Thus, we

are reminded that the process of solving a typical algebra

problem probably involves not only the recall of an appropriate

schema but also the construction of an initial problem

represen-tation The representation provides a framewcrk 6) which the solver

can zipply the retrieved schema

It is not at all uncommon to find first-year algebra students

who can solve a problem when it matches exactly, the "mo'clel" problem

they have already solved but who cannot solve a similar problem that

they perceive as different One reasonable explanation for such'behavior

is the absence of semantic processing of problem information. In

other words, the students may be searching his memory for a "model"

problem to apply to the given situation and failing to find a "match".The failure may be due to the non-existence of an appropriate schema

or the misdirection of the search due to the student's lack of problemrepresentation to guide the search ,

Construction of a meaningful problem representation.involves the

incorporation of semantic knowledge in the problem understanding

process The work of Larkin (Note 10) and Heller and Greeno (Note 5)

discussed earlier suggest the critical importance of semantic

processing in successful problem-solving performance. Further

sup-port for this view may be found in the work of Paige and Simon

(1966) who reported that solvers who used a direct translation

approach to solving problems containing containing

dictory information were able to obtain "impossible" solutions

and not perceive the contradiction. They found that subjrcts

who constructed "auxiliary representations" of tho ptoblem situation

(e.g drawings) or who relied on semantic, substantive information

in the solution process were considerably more successful at recogrizingthe presence of incongruities in the problem's conditions Krutetskii(1976) als9 reported similar findings in his work with highly capablemathematics students The findings of the studies reported in this

section strongly suggest that future tesearch pay specific attention

to the mechanisms of schema construction and problem representation

formulation

Another focus for further research might be the nature of

schema composition; i.e what knowledge is embedded in one's

problem schema?' It seems reasonable to expect that successful

problem solvers may exhibit certain process similarities, such

as those discussed by Larkin (Note 10), but that they may possess

different knowledge structures For example, two solvers may

be quite successful in solving typical Distance/Rate/Time

problems, yet they may have different schemata for such problems.

One solver might view these problems as being similar to other

typical alge:ita problems, such as "mixture" and "coin" problems,

since they all involve the general structural notian:

Total = Rate Per Unit x Number of Units

Another solver's schema might include specific details regarding

the assumptions of such problems; for example, uniform rate of trasiel,smoothness of surface, diversity oE path, and instantaneous "turn

around" Another solver might not have these details explicitly

N\

stated, but may operate with "default" valuehat are equivalent

to the necessary assumptions

In addition to the few examples given above, it is clearly

possible to propose other possible individual differences in schema

composition If such differences do exist, it may be fruitful for

researchers to examine not only the expert-novice distinctions

that have captured our attention for the past decade, but also

expert-expert and novice-novice distinctions with respect to

processes and with respect to schema composition. By pursuing this

line of research, we may learn if there are necessary and sufficient

components of problem schemata for various classes of problems, and

thi s information could be useful in guiding instruction.

Of course, not all problem solving behavior can be neatly

described in terms of schemata When subjects have little or no

experience in solving a class of problems, the usefulness of schemata

Is limited When solving a new problem, a successful problem solver

presumably uses information, pro,:edures, and more general nations

that have been obtained ftom previous experience and training. As

noted earlier in this paper, Gestalt psychologists have demonstrated

that prior experience may have a negative effect in problem solving

.,such as the Tower of Hanoi or the Missionaries-Cannibals problem.

The classic study by Reed, Ernst, and Banerji (1974) suggested thatpositWe transfer occurred only when subjects were told of the

relationship between the problems and only when they solved the

more difficult problem of the pair first. Kulm and Days (1979)

used an information-theoretic approach to study transfer between

problems with related structures. They reported that the solution

of related problems appeared to help subjects focus on relevant

strategies, but that different problem contexts appeared to

interfere with transfer

Silver (Note 12) has suggested that the potential transfer to

a new problem is greatly influenced by the solver's initial perception

of the problem's relationships to previously solved problems;

furthermore, the initial perception is largely a function of what

aspects of a problem the solver views to ize mathematically relevant

to its solution In other words, the solver must not only recognizethat the new problem is related to previously encountered problemsbut also identify the important mathematical considerations that arerelevant to the relationship with previous problems. Of course thesolver must also have the necessary information stored in long term

memory

The question of what gets remembered after a problem solutionepisode has been dealt with at length by Reed andJohnsen (1977)

and to a lesser extent by Jacoby (1978). Unfortunately, the

literature on this subject is sketchy and largely based on

non-mathematical problems In the next section, we will discuss the

few studies that have dealt specifically with long-term retention

of mathematical problems.

Individual Differences in Memoiy and Problem Solving

In studying individual differences in technical problem solving,

many researchers have examined the differential processing characteristics

of novices and experts (e.g Chi & Glaser, Note 14; Larkin, Note 10,

Simon & Simon, 1978) The data from these studies generally suggest

that experts are capable of deeply processing problem information

very early in the solution process, thus facilitating solution plan

formulation for complex problems and essentially solving "immediately"simple problems

Since, as Miller, Galanter, and Pribram (1960) have noted, the

major source of new plans is old plans, the process differences noted

early in the solution are likely indicators of differences in the

memories of experts and novices In fact, the classic work of de Groot(1966) on the memories of skilled and unskilled chess players has

stimulated much of the research into expert-novice distinctions The

,data from de Groot's study and subsequent studies (Chase & Simon 1973a,1973h; Frey & Adesman, 1976) demonstrated that skilled chess players'were considerably more successful than weat$er players at reproducing

meaningful chess situations, and that the results were not attributable

to superior memory or hetter guessing on the part of the experts.

Individual differences in memory associated with mathematical

problem solving is a largely unexplored area Krutetskii (1976) notedthat skillful problem solvers were able to recall accurately the

structure of a mathematics.problem even after long periods of time;

whereas, poor problem solvers tended to recall, if anything, only thedetails of the problem's statement.

Recently, Silver (1977, Note 12, Note 13) has r.?Torted data

9.-'t suggesting that good and poor problem solvers demonstrate qualitativedifferences in their recall of problem information and in their

perception of problem relatedness Regarding the latter, Silver (1977)had students sort a set of word problems into groups that were

"mathematically related" The data indicated that good problem solverstended to group the problems on the basis of mathematical structure,

even,when they lacked specific techniques designed to solve problems with the given structure To examine differences in recall, Silver

(Note 12, Note 13) asked stUdents to reproduce all they could rememberabout a mathematical problem and its solution. Recall was examined

on several occasions, both before and after presentation of problem

solutions, and the data indicated superior structurtl recall by

skillful prablem solvers Furthermore, the data indicated that

skillful problem solvers were better able to transfer information

from one problem solution to the solution of a structurally related

problem (Silver, Note 12) and that skillful problem solvers tended.

to cluster related information from several problems in terms of

problem structure, whereas, less skilled solvers tended not to

cluster or to cluster in terms of problem details or cover story

(Silver, Note 13)

Much more attention is neded to the issue of individual differences

in mathematical problem-solving performance that may be related to

memory As Hunt (1978) has remarked, "Individual differences are

undoubtedly due both to differences in peoples' mental machinery

and to differences in how they program that machinery to bring it

to bear upon the problems they face."

6; 17

Salving Word Problems: A Final Word

Word problems have been the subject of much research .

activity by psychologists and mathematics educators Since

: 'word prOblems require the solver to read and understand a

written.passage, to select and apply mathematical principles,

algorithms or procAdures in determining the value of one or

more unknown quantities, and to interpret the mathemacical

,

solution with respect to the verbal information given in the

problem, they represent a poillt Of intersectian of the concerns

of those interested,in mathematical competence and those

interested in prose text compreheniian Thus it is fitting

that some of the maj'or conclusions of this review parallel

results found in the literature on prose text comprehension.

For example, the influence and power of schemata in guiding encodimand retrieval of text information has been demonstrated by Anderson,

Reynolds, Schallert, and Goetz (1977) and Mandler and Johnson (1977).Another parallel finding is the existence of differences between goodand poor readers' recall of thematically relevant material (Smiley,

Oakley, Worthen, Campione, & Brown, Note 2).

The major conclusions of this review are that the criticalprocesses in mathematical word problem solving involve the solver

in constructing an accurate representation of the problem and using

that representation as a guide in recalling relevant and necessary

information, often in the form of schemata, to solve the problem

We have seen that skilled and unskilled solvers demonstrate

qualitative differences in the representations they

construct and the structures from which they retrieve needed

information Nevertheless, we have also seen that our knowledge of howmemory is involved in mathematical problem solving is very incomplete.Perhaps this review has sharpened a few questions for further

study

9

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3 4